Correlation

1 / 24

# Correlation - PowerPoint PPT Presentation

Correlation. Review and Extension. Questions to be asked…. Is there a linear relationship between x and y? What is the strength of this relationship? Pearson Product Moment Correlation Coefficient (r) Can we describe this relationship and use this to predict y from x? y=bx+a

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Correlation' - Audrey

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Correlation

Review and Extension

Questions to be asked…
• Is there a linear relationship between x and y?
• What is the strength of this relationship?
• Pearson Product Moment Correlation Coefficient (r)
• Can we describe this relationship and use this to predict y from x?
• y=bx+a
• Is the relationship we have described statistically significant?
• Not a very interesting one if tested against a null of r = 0
Other stuff
• Check scatterplots to see whether a Pearson r makes sense
• Use both r and r2 to understand the situation
• If data is non-metric or non-normal, use “non-parametric” correlations
• Correlation does not prove causation
• True relationship may be in opposite direction, co-causal, or due to other variables
• However, correlation is the primary statistic used in making an assessment of causality
• ‘Potential’ Causation
Possible outcomes
• -1 to +1
• As one variable increases/decreases, the other variable increases/decreases
• Positive covariance
• As one variable increases/decreases, another decreases/increases
• Negative covariance
• No relationship (independence)
• r = 0
• Non-linear relationship
Covariance
• The variance shared by two variables
• When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg)
• cov (x,y) = pos.
• When X and Y move in opposite directions
• cov (x,y) = neg.
• When no constant relationship
• cov (x,y) = 0
Covariance is not very meaningful on its own and cannot be compared across different scales of measurement
• Solution: standardize this measure
• Pearson’s r:
Factors affecting Pearson r
• Linearity
• Heterogeneous subsamples
• Range restrictions
• Outliers
Linearity
• Nonlinear relationships will have an adverse effect on a measure designed to find a linear relationship
Heterogeneous subsamples
• Sub-samples may artificially increase or decrease overall r.
• Solution - calculate r separately for sub-samples & overall, look for differences
Range restriction
• Limiting the variability of your data can in turn limit the possibility for covariability between two variables, thus attenuating r.
• Common example occurs with Likert scales
• E.g. 1 - 4 vs. 1 - 9
• However it is also the case that restricting the range can actually increase r if by doing so, highly influential data points would be kept out
Effect of Outliers
• Outliers can artificially and dramatically increase or decrease r
• Options
• Compute r with and without outliers
• Conduct robustified R!
• For example, recode outliers as having more conservative scores (winsorize)
• Transform variables
What else?
• r is the starting point for any regression and related method
• Both the slope and magnitude of residuals are reflective of r
• R = 0 slope =0
• As such a lone r doesn’t really provide much more than a starting point for understanding the relationship between two variables
Robust Approaches to Correlation
• Rank approaches
• Winsorized
• Percentage Bend
Rank approaches: Spearman’s rho and Kendall’s tau
• Spearman’s rho is calculated using the same formula as Pearson’s r, but when variables are in the form of ranks
• Simply rank the data available
• X = 10 15 5 35 25 becomes
• X = 2 3 1 5 4
• Do this for X and Y and calculate r as normal
• Kendall’s tau is a another rank based approach but the details of its calculation are different
• For theoretical reasons it may be preferable to Spearman’s, but both should be consistent for the most part and perform better than Pearson’s r when dealing with non-normal data
Winsorized Correlation
• As mentioned before, Winsorizing data involves changing some decided upon percentage of extreme scores to the value of the most extreme score (high and low) which is not Winsorized
• X = 1 2 3 4 5 6 becomes
• X = 2 2 3 4 5 5
• Winsorize both X and Y values (without regard to each other) and compute Pearson’s r
• This has the advantage over rank-based approaches since the nature of the scales of measurement remain unchanged
• For theoretical reasons (recall some of our earlier discussion regarding the standard error for trimmed means) a Winsorized correlation would be preferable to trimming
• Though trimming is preferable for group comparisons
Methods Related to M-estimators
• The percentage bend correlation utilizes the median and a generalization of MAD
• A criticism of the Winsorized correlation is that the amount of Winsorizing is fixed in advance rather than determined by the data, and the rpb gets around that
• While the details can get a bit technical, you can get some sense of what is going on by relying on what you know regarding the robust approach in general
• With independent X and Y variables, the values of robust approaches to correlation will match the Pearson r
• With nonnormal data, the robust approaches described guard against outliers on the respective X and Y variables while Pearson’s r does not
Problem
• While these alternative methods help us in some sense, an issue remains
• When dealing with correlation, we are not considering the variables in isolation
• Outliers on one or the other variable, might not be a bivariate outlier
• Conversely what might be a bivariate outlier may not contain values that are outliers for X or Y themselves
Global measures of association
• Measures are available that take into account the bivariate nature of the situation
• Minimum Volume Ellipsoid Estimator (MVE)
• Minimum Covariance Determinant Estimator (MCD)
Minimum Volume Ellipsoid Estimator
• Robust elliptic plot (relplot)
• Relplots are like scatterplot boxplots for our data where the inner circle contains half the values and anything outside the dotted circle would be considered an outlier
• A strategy for robust estimation of correlation would be to find the ellipse with the smallest area that contains half the data points
• Those points are then used to calculate the correlation
• The MVE
Minimum Covariance Determinant Estimator
• The MCD is another alternative we might used and involves the notion of a generalized variance, which is a measure of the overall variability among a cloud of points
• For the more adventurous, see my /6810 page for info matrices and their determinants
• The determinant of a matrix is the generalized variance
• For the two variable situation
• As we can see, as r is a measure of linear association, the more tightly the points are packed the larger it would be, and subsequently smaller the generalized variance would be
• The MCD picks that half of the data which produces the smallest generalized variance, and calculates r from that
Global measures of association
• Note that both the MVE and MCD can be extended to situations with more than two variables
• We’d just be dealing with a larger matrix
• Example using the Robust library in S-Plus
• OMG! Drop down menus even!
Remaining issues: Curvature
• The fact is that straight lines may not capture the true story
• We may often fail to find noticeable relationships because our r, whichever method of “Pearsonesque” one we choose, is trying to specify a linear relationship
• There may still be a relationship, and a strong one, just more complex
Summary
• Correlation, in terms of Pearson r, gives us a sense of the strength of a linear association between two variables
• One data point can render it a useless measure, as it is not robust to outliers
• Measures which are robust are available, and some take into account the bivariate nature of the data
• However, curvilinear relationships may exist, and we should examine the data to see if alternative explanations are viable